Certain Norm Related Quantities of Unbounded Linear Operators

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

Spectral methods for linear inverse problems with unbounded operators

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above.

Several approaches are known to generalize the shift operator in the complex domain. Specific generalization procedures lead to significantly different operators of the shift type even in one specific area of complex analysis. In this paper, a certain general operator of the shift type is defined. The properties of this operator are studied and the question of the continuity of this definition is considered. The properties of a certain operator to a certain extent re-peat the properties of the previously studied operators of the shift type. This allows us to speak of the possibility of extending the main results on spectral synthesis in the complex domain to convolutional equations of a more general form.

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space X is reflexive.The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.o.Introduction.One of the principal results of this paper is that each scalar-type spectral operator T in the quasicomplete, locally convex space X is reflexive, i.e. the strongly closed subalgebra generated by the identity and T in .P(X), the space of continuous linear operators on X, consists precisely of those continuous linear operators on X which leave invariant each (closed) T-invariant subspace of X.For the case that X is a Banach space, this result was established by Gillespie [9] via an interesting factorization theorem in Banach function spaces, a method which does not appear to extend readily to the more general setting.The present approach, however, avoids factorization theorems by showing directly that each continuous linear functional on the strongly closed algebra generated by a Bade-complete, equicontinuous Boolean algebra of projections in X has a representation of the form < .x, x') for some x E X and x' E X', where X' denotes the dual space of X, a result which goes back to R. Pallu de la Barriere [20] for the case of Abelian von Neumann algebras in Hilbert space.Our method is based on ideas from the theory of Riesz spaces and yields considerable simplification of technique, even in the setting of Banach spaces.The cornerstone of the present paper is the extension of the reflexivity theorem of Bade [2] to the setting of locally convex spaces proved in [6] via the theory of closed spectral measures and further refined and sharpened in [5] using purely intrinsic methods, based on order considerations.One of the new features which emerged from the approach of [5] was a type of "automatic continuity" theorem for a certain class of everywhere defined linear operators, even in the absence of a suitable closed-graph theorem.This idea is exploited in 1 to show that an everywhere

In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [1].

Nonlinear random equations involving operators of monotone type

Algebraic Studies of Symmetric Operators By Zhiqin Shi Dissertation Director: Professor William Keigher There was an old problem of G. C. Rota regarding the classification of all linear operators on associative algebras that satisfy algebraic identities. We only know very few of such operators at the beginning, for example, the derivative operator, average operator, difference operator and Rota-Baxter operator. Recently L. Guo, W. Sit and R. Zhang revisited Rota’s problem in a paper by concentrating on two classes of operators: differential type operators and Rota-Baxter type operators. One of the Rota-Baxter type operators they found is the symmetric Rota-Baxter operator which symmetrizes the RotaBaxter operator. In this dissertation, we initiate a systematic study of the symmetric Rota-Baxter operator, extending the previous works on the original Rota-Baxter operator. After giving basic properties and examples, we construct free symmetric Rota-Baxter algebras on an algebra and on a set by bracketed words and rooted trees separately. We then use the free symmetric Rota-Baxter algebra to obtain an extension of the well known dendriform algebra and its free objects. Finally, we extend our study to differential algebras. We construct the free symmetric differential Rota-Baxter algebra based on the previous free symmetric Rota-Baxter algebra on a set and the free symmetric differential algebra.

: The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

Similar to the property of a linear Calderon-Zygmund operator, a linear fractional type operator Iα associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ⩽ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b, Iα] is continuous from an atomic Hardy space Hbp into Lp, where Hbp is a subspace of the Hardy space Hp for n/(n+1) < p ⩽ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint \((H_{b_1 }^{p_1 } \times \cdots \times H_{bm}^{p_m } ,L^p )\) boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderon-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ (Lipβ)m(ℝn).

The algorithm of application of the method of steepest descent to the solution of problems of structural mechanics and solid mechanics, described by nonlinear differential equations. For application of this method to nonlinear operators are described by a sequence of linear operators in incremental form, unlimited and complex linear operator, in line with the idea of L. V. Kantorovich is limited to a simple linear unbounded operator.

In this paper we study point-to-point Dirichlet problem for equation of oscillator motion with viscoelastic damping for damping order 1 < α < 2. It was shown, that operator, generated by this problem, is dissipative operator of Keldysh type with oscillation properties. The authors of the work proved that the linear operator accompanying a mechanical system in which there is energy dissipation is dissipative. The article it follows that the linear operator is oscillatory and since the linear operator describes the motion of the oscillator, this operator has a complex of oscillatory properties. In addition, it is shown that in the case when the 0 < 0 < 1 linear operator accompanying point-to-point Dirichlet problem has the above properties, and the authors have determined the eigenfunctions of problem point-to-point Dirichlet and it is proved that the system of functions of point-to-point Dirichlet problem forms a basis in L2 (0,1).

In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the Lp-Lq-boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the Lp-estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate Lp-boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove L∞-BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle Lp-Lq boundedness of the Fourier multipliers and pseudo-multipliers operators for the range 1<p≤2≤q<∞. On the way to achieve our goal of Lp-Lq boundedness, we prove two classical inequalities, namely, Paley inequality and Hausdorff–Young–Paley inequality for smooth manifolds. Finally, we present some examples about the well-posedness of abstract non-linear equations.

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP [Formula: see text] that is the image of another GP [Formula: see text] under a linear transformation [Formula: see text] acting on the sample paths of [Formula: see text] are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when [Formula: see text] is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator [Formula: see text] acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.

Sufficient conditions for the local null controllability of non-linear functional differential systems with unbounded linear operators in Banach space are established. The results are obtained using the semigroup of linear operators, fractional powers of operators, and the Schauder fixed-point theorem. Applications to parabolic differential systems are given.